Evaluation Performance of 2nd Order Nonlinear System: Baseline Control Tunable Gain Sliding Mode Methodology

Farzin Piltan, Javad Meigolinedjad, Saleh Mehrara & Sajad Rahmdel
International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (3) : 2012 192
Evaluation Performance of 2nd
Order Nonlinear System:
Baseline Control Tunable Gain Sliding Mode Methodology
Farzin Piltan SSP.ROBOTIC@yahoo.com
Industrial Electrical and Electronic
Engineering SanatkadeheSabze
Pasargad. CO (S.S.P. Co), NO:16
,PO.Code 71347-66773, Fourth floor
Dena Apr , Seven Tir Ave , Shiraz , Iran
Javad Meigolinedjad SSP.ROBOTIC@yahoo.com
Saleh Mehrara SSP.ROBOTIC@yahoo.com
Sajad Rahmdel SSP.ROBOTIC@yahoo.com
Abstract
Refer to this research, a baseline error-based tuning sliding mode controller (LTSMC) is
proposed for robot manipulator. Sliding mode controller (SMC) is an important nonlinear controller
in a partly uncertain dynamic system’s parameters. Sliding mode controller has difficulty in
handling unstructured model uncertainties. It is possible to solve this problem by combining
sliding mode controller and adaption law which this method can helps improve the system’s
tracking performance by online tuning method. Since the sliding surface gain (λ) is adjusted by
baseline tuning method, it is continuous. In this research new λ is obtained by the previous λ
multiple sliding surface slopes updating factor ሺδሻ. Baseline error-based tuning sliding mode
controller is stable model-based controller which eliminates the chattering phenomenon without to
use the boundary layer saturation function. Lyapunov stability is proved in baseline error-based
tuning sliding mode controller based on switching (sign) function. This controller has acceptable
performance in presence of uncertainty (e.g., overshoot=0%, rise time=0.4 second, steady state
error = 1.8e-10 and RMS error=1.16e-12).
Keywords: Baseline Error-based Tuning Sliding Mode Controller, Sliding Mode Controller,
Unstructured Model Uncertainties, Adaptive Method, Sliding Surface Gain, Sliding Surface Slopes
Updating Factor, Chattering Phenomenon.

1. INTRODUCTION, BACKGROUND and MOTIVATION
Introduction
Controller is a device which can sense information from linear or nonlinear system (e.g., robot
manipulator) to improve the systems performance [1-4]. The main targets in designing control
systems are stability, good disturbance rejection, and small tracking error[5-6]. Several industrial
robot manipulators are controlled by linear methodologies (e.g., Proportional-Derivative (PD)
controller, Proportional- Integral (PI) controller or Proportional- Integral-Derivative (PID)
controller), but when robot manipulator works with various payloads and have uncertainty in
dynamic models this technique has limitations. From the control point of view, uncertainty is
divided into two main groups: uncertainty in unstructured inputs (e.g., noise, disturbance) and
uncertainty in structure dynamics (e.g., payload, parameter variations). In some applications robot
manipulators are used in an unknown and unstructured environment, therefore strong
mathematical tools used in new control methodologies to design nonlinear robust controller with
an acceptable performance (e.g., minimum error, good trajectory, disturbance rejection). Sliding
mode controller is a powerful nonlinear robust controller under condition of partly uncertain
dynamic parameters of system [7]. This controller is used to control of highly nonlinear systems
especially for robot manipulators. Chattering phenomenon in uncertain dynamic parameter is the
main drawback in pure sliding mode controller [8-20]. The chattering phenomenon problem in
pure sliding mode controller is reduced by using linear saturation boundary layer function but
prove the stability is very difficult. In various dynamic parameters systems that need to be training
on-line adaptive control methodology is used. Adaptive control methodology can be classified into
two main groups, namely, traditional adaptive method and fuzzy adaptive method [41-70]. Fuzzy
adaptive method is used in systems which want to training parameters by expert knowledge [21-
38]. Traditional adaptive method is used in systems which some dynamic parameters are known.
In this research in order to solve disturbance rejection and uncertainty dynamic parameter,
adaptive method is applied to artificial sliding mode controller [39-40, 71-77].
Robot manipulator is a collection of links that connect to each other by joints, these joints can be
revolute and prismatic that revolute joint has rotary motion around an axis and prismatic joint has
linear motion around an axis. Each joint provides one or more degrees of freedom (DOF). From
the mechanical point of view, robot manipulator is divided into two main groups, which called;
serial robot links and parallel robot links. In serial robot manipulator, links and joints is serially
connected between base and final frame (end-effector) [15-25]. Parallel robot manipulators have
many legs with some links and, where in these robot manipulators base frame has connected to
the final frame. Most of industrial robots are serial links, which in ݊ degrees of freedom serial link
robot manipulator the axis of the first three joints has a known as major axis, these axes show the
position of end-effector, the axis number four to six are the minor axes that use to calculate the
orientation of end-effector and the axis number seven to ݊ use to reach the avoid the difficult
conditions (e.g., surgical robot and space robot manipulator). Kinematics is an important subject
to find the relationship between rigid bodies (e.g., position and orientation) and end-effector in
robot manipulator. The mentioned topic is very important to describe the three areas in robot
manipulator: practical application such as trajectory planning, essential prerequisite for some
dynamic description such as Newton’s equation for motion of point mass, and control purposed
therefore kinematics play important role to design accurate controller for robot manipulators.
Robot manipulator kinematics is divided into two main groups: forward kinematics and inverse
kinematics where forward kinematics is used to calculate the position and orientation of end-
effector with given joint parameters (e.g., joint angles and joint displacement) and the activated
position and orientation of end-effector calculate the joint variables in Inverse Kinematics[1-6].
Dynamic modeling of robot manipulators is used to describe the behavior of robot manipulator
such as linear or nonlinear dynamic behavior, design of model based controller such as pure
sliding mode controller and pure computed torque controller which design these controller are
based on nonlinear dynamic equations, and for simulation. The dynamic modeling describes the
relationship between joint motion, velocity, and accelerations to force/torque or current/voltage
and also it can be used to describe the particular dynamic effects (e.g., inertia, coriolios,
centrifugal, and the other parameters) to behavior of system[1-15].

Background
Chattering phenomenon can causes some problems such as saturation and heats the
mechanical parts of robot arm or drivers. To reduce or eliminate the oscillation, various papers
have been reported by many researchers which one of the best method is; boundary layer
saturation method [1]. In boundary layer linear saturation method, the basic idea is the
discontinuous method replacement by linear continuous saturation method with small
neighborhood of the switching surface. This replacement caused to considerable chattering
reduction. Slotine and Sastry have introduced boundary layer method instead of discontinuous
method to reduce the chattering[21]. Slotine has presented sliding mode controller with boundary
layer to improve the industry application [22]. Palm has presented a fuzzy method to nonlinear
approximation instead of linear approximation inside the boundary layer to improve the chattering
and control the result performance[23]. Moreover, Weng and Yu improved the previous method
by using a new method in fuzzy nonlinear approximation inside the boundary layer and adaptive
method[24]. In various dynamic parameters systems that need to be training on-line, adaptive
control methodology is used. Mathematical model free adaptive method is used in systems which
want to training parameters by performance knowledge. In this research in order to solve
disturbance rejection and uncertainty dynamic parameter, adaptive method is applied to sliding
mode controller. Mohan and Bhanot [40] have addressed comparative study between some
adaptive fuzzy, and a new hybrid fuzzy control algorithm for robot arm control. They found that
self-organizing fuzzy logic controller and proposed hybrid integrator fuzzy give the best
performance as well as simple structure. Temeltas [46] has proposed fuzzy adaption techniques
for VSC to achieve robust tracking of nonlinear systems and solves the chattering problem.
Conversely system’s performance is better than sliding mode controller; it is depended on
nonlinear dynamic equqation. Hwang et al. [47]have proposed a Tagaki-Sugeno (TS) fuzzy model
based sliding mode controller based on N fuzzy based linear state-space to estimate the
uncertainties. A MIMO FVSC reduces the chattering phenomenon and reconstructs the
approximate the unknown system has been presented for a nonlinear system [42]. Yoo and Ham
[58]have proposed a MIMO fuzzy system to help the compensation and estimation the torque
coupling. This method can only tune the consequence part of the fuzzy rules. Medhafer et al. [59]
have proposed an indirect adaptive fuzzy sliding mode controller to control nonlinear system. This
MIMO algorithm, applies to estimate the nonlinear dynamic parameters. Compared with the
previous algorithm the numbers of fuzzy rules have reduced by introducing the sliding surface as
inputs of fuzzy systems. Guo and Woo [60]have proposed a SISO fuzzy system compensate and
reduce the chattering. Lin and Hsu [61] can tune both systems by fuzzy rules. Eksin et. al [70]
have designed mathematical model-free sliding surface slope in fuzzy sliding mode controller.
This paper is organized as follows. In section 2, main subject of sliding mode controller, proof of
stability and dynamic formulation of robot manipulator are presented. This section covered the
following details, classical sliding mode control, classical sliding for robotic manipulators, proof of
stability in pure sliding mode controller, chatter free sliding controller and nonlinear dynamic
formulation of system. A methodology of proposed method is presented in section 3, which
covered the baseline tuning error-based tuning sliding mode controller and proofs the stability in
this method and applied to robot manipulator. In section 4, the sliding mode controller and
proposed methodology are compared and discussed. In section 5, the conclusion is presented.
2. THEOREM: DYNAMIC FORMULATION OF ROBOTIC MANIPULATOR,
SLIDING MODE FORMULATION APPLIED TO ROBOT ARM, PROOF OF
STABILITY
Dynamic of robot arm: The equation of an n-DOF robot manipulator governed by the following
equation [1, 4, 15-29, 63-70]:
ࡹሺࢗሻࢗሷ ൅ ࡺሺࢗ, ࢗሶ ሻ ൌ ࣎ (1)
Where τ is actuation torque, M (q) is a symmetric and positive define inertia matrix, ܰሺ‫,ݍ‬ ‫ݍ‬ሶሻ is the
vector of nonlinearity term. This robot manipulator dynamic equation can also be written in a
following form [1-29]:

࣎ ൌ ࡹሺࢗሻࢗሷ ൅ ࡮ሺࢗሻሾࢗሶ ࢗሶ ሿ ൅ ࡯ሺࢗሻሾࢗሶ ሿ૛
൅ ࡳሺࢗሻ (2)
Where B(q) is the matrix of coriolios torques, C(q) is the matrix of centrifugal torques, and G(q) is
the vector of gravity force. The dynamic terms in equation (2) are only manipulator position. This
is a decoupled system with simple second order linear differential dynamics. In other words, the
component ‫ݍ‬ሷ influences, with a double integrator relationship, only the joint variable‫ݍ‬௜,
independently of the motion of the other joints. Therefore, the angular acceleration is found as to
be [3, 41-62]:
ࢗሷ ൌ ࡹି૚ሺࢗሻ. ሼ࣎ െ ࡺሺࢗ, ࢗሶ ሻሽ (3)
This technique is very attractive from a control point of view.
Sliding Mode methodology: Consider a nonlinear single input dynamic system is defined by [6]:
࢞ሺ࢔ሻ
ൌ ࢌሺ࢞ሬሬԦሻ ൅ ࢈ሺ࢞ሬሬԦሻ࢛ (4)
Where u is the vector of control input, ࢞ሺ࢔ሻ
is the ࢔࢚ࢎ
derivation of ࢞, ࢞ ൌ ሾ࢞, ࢞ሶ, ࢞ሷ, … , ࢞ሺ࢔ି૚ሻ
ሿࢀ
is the
state vector, ࢌሺ࢞ሻ is unknown or uncertainty, and ࢈ሺ࢞ሻ is of known sign function. The main goal to
design this controller is train to the desired state; ࢞ࢊ ൌ ሾ࢞ࢊ, ࢞ሶࢊ, ࢞ሷࢊ, … , ࢞ࢊ
ሺ࢔ି૚ሻ
ሿࢀ
, and trucking
error vector is defined by [6]:
࢞෥ ൌ ࢞ െ ࢞ࢊ ൌ ሾ࢞෥, … , ࢞෥ሺ࢔ି૚ሻ
ሿࢀ (5)
A time-varying sliding surface ࢙ሺ࢞, ࢚ሻ in the state space ࡾ࢔
is given by [6]:
࢙ሺ࢞, ࢚ሻ ൌ ሺ
ࢊ
ࢊ࢚
൅ ࣅሻ࢔ି૚
࢞෥ ൌ ૙
(6)
where λ is the positive constant. To further penalize tracking error, integral part can be used in
sliding surface part as follows [6]:
࢙ሺ࢞, ࢚ሻ ൌ ሺ
ࢊ
ࢊ࢚
൅ ࣅሻ࢔ି૚ ቆන ࢞෥
࢚
૙
ࢊ࢚ቇ ൌ ૙
(7)
The main target in this methodology is kept the sliding surface slope ࢙ሺ࢞, ࢚ሻ near to the zero.
Therefore, one of the common strategies is to find input ࢁ outside of ࢙ሺ࢞, ࢚ሻ [6].
૚
૛
ࢊ
ࢊ࢚
࢙૛ሺ࢞, ࢚ሻ ൑ െࣀ|࢙ሺ࢞, ࢚ሻ|
(8)
where ζ is positive constant.
If S(0)>0՜
ࢊ
ࢊ࢚
ࡿሺ࢚ሻ ൑ െࣀ (9)
To eliminate the derivative term, it is used an integral term from t=0 to t=࢚࢘ࢋࢇࢉࢎ
න
ࢊ
ࢊ࢚
࢚ୀ࢚࢘ࢋࢇࢉࢎ
࢚ୀ૙
ࡿሺ࢚ሻ ൑ െ න ࣁ ՜ ࡿ
࢚ୀ࢚࢘ࢋࢇࢉࢎ
࢚ୀ૙
ሺ࢚࢘ࢋࢇࢉࢎሻ െ ࡿሺ૙ሻ ൑ െࣀሺ࢚࢘ࢋࢇࢉࢎ െ ૙ሻ
(10)
Where ‫ݐ‬௥௘௔௖௛ is the time that trajectories reach to the sliding surface so, suppose S(‫ݐ‬௥௘௔௖௛ ൌ 0ሻ
defined as
૙ െ ࡿሺ૙ሻ ൑ െࣁሺ࢚࢘ࢋࢇࢉࢎሻ ՜ ࢚࢘ࢋࢇࢉࢎ ൑
ࡿሺ૙ሻ
ࣀ
(11)
and
࢏ࢌ ࡿሺ૙ሻ ൏ 0 ՜ ૙ െ ࡿሺ૙ሻ ൑ െࣁሺ࢚࢘ࢋࢇࢉࢎሻ ՜ ࡿሺ૙ሻ ൑ െࣀሺ࢚࢘ࢋࢇࢉࢎሻ ՜ ࢚࢘ࢋࢇࢉࢎ ൑
|ࡿሺ૙ሻ|
ࣁ
(12)
Equation (12) guarantees time to reach the sliding surface is smaller than
|ࡿሺ૙ሻ|
ࣀ
since the
trajectories are outside of ܵሺ‫ݐ‬ሻ.
࢏ࢌ ࡿ࢚࢘ࢋࢇࢉࢎ
ൌ ࡿሺ૙ሻ ՜ ࢋ࢘࢘࢕࢘ሺ࢞ െ ࢞ࢊሻ ൌ ૙ (13)
suppose S is defined as
࢙ሺ࢞, ࢚ሻ ൌ ሺ
ࢊ
ࢊ࢚
൅ ࣅሻ ࢞෥ ൌ ሺ࢞ሶ െ ࢞ሶࢊሻ ൅ ࣅሺ࢞ െ ࢞ࢊሻ
(14)
The derivation of S, namely, ܵሶ can be calculated as the following;

‫܁‬ሶ ൌ ሺ‫ܠ‬ሷ െ ‫ܠ‬ሷ‫܌‬ሻ ൅ ૃሺ‫ܠ‬ሶ െ ‫ܠ‬ሶ‫܌‬ሻ (15)
suppose the second order system is defined as;
‫ܠ‬ሷ ൌ ܎ ൅ ‫ܝ‬ ՜ ‫܁‬ሶ ൌ ܎ ൅ ‫܃‬ െ ‫ܠ‬ሷ‫܌‬ ൅ ૃሺ‫ܠ‬ሶ െ ‫ܠ‬ሶ‫܌‬ሻ (16)
Where ࢌ is the dynamic uncertain, and also since ܵ ൌ 0 ܽ݊݀ ܵሶ ൌ 0, to have the best
approximation ,ࢁ෡ is defined as
ࢁ෡ ൌ െࢌ෠ ൅ ࢞ሷࢊ െ ࣅሺ‫ܠ‬ሶ െ ‫ܠ‬ሶ‫܌‬ሻ (17)
A simple solution to get the sliding condition when the dynamic parameters have uncertainty is
the switching control law:
ࢁࢊ࢏࢙ ൌ ࢁ෡ െ ࡷሺ࢞ሬሬԦ, ࢚ሻ · ‫ܖ܏ܛ‬ሺ࢙ሻ (18)
where the switching function ‫ܖ܏ܛ‬ሺ‫܁‬ሻ is defined as [1, 6]
࢙ࢍ࢔ሺ࢙ሻ ൌ ൝
૚ ࢙ ൐ 0
െ૚ ࢙ ൏ 0
૙ ࢙ ൌ ૙
(19)
and the ࡷሺ࢞ሬሬԦ, ࢚ሻ is the positive constant. Suppose by (8) the following equation can be written as,
૚
૛
ࢊ
ࢊ࢚
࢙૛ሺ࢞, ࢚ሻ ൌ ‫܁‬ ·ሶ ‫܁‬ ൌ ൣࢌ െ ࢌ෠ െ ࡷ‫ܖ܏ܛ‬ሺ࢙ሻ൧ · ࡿ ൌ ൫ࢌ െ ࢌ෠൯ · ࡿ െ ࡷ|ࡿ|
(20)
and if the equation (12) instead of (11) the sliding surface can be calculated as
࢙ሺ࢞, ࢚ሻ ൌ ሺ
ࢊ
ࢊ࢚
൅ ࣅሻ૛ ቆන ࢞෥
࢚
૙
ࢊ࢚ቇ ൌ ሺ࢞ሶ െ ࢞ሶࢊሻ ൅ ૛ࣅሺ࢞ሶ െ ࢞ሶࢊሻ െ ࣅ૛ሺ࢞ െ ࢞ࢊሻ
(21)
in this method the approximation of ࢁ is computed as [6]
ࢁ෡ ൌ െࢌ෠ ൅ ࢞ሷࢊ െ ૛ࣅሺ‫ܠ‬ሶ െ ‫ܠ‬ሶ‫܌‬ሻ ൅ ૃ૛ሺ‫ܠ‬ െ ‫ܠ‬‫܌‬ሻ (22)
Based on above discussion, the sliding mode control law for a multi degrees of freedom robot
manipulator is written as [1, 6]:
࣎ ൌ ࣎ࢋࢗ ൅ ࣎ࢊ࢏࢙ (23)
Where, the model-based component ࣎ࢋࢗ is the nominal dynamics of systems and ࣎ࢋࢗ for first 3
DOF PUMA robot manipulator can be calculate as follows [1]:
࣎ࢋࢗ ൌ ൣࡹି૚ሺ࡮ ൅ ࡯ ൅ ࡳሻ ൅ ࡿሶ൧ࡹ (24)
and ࣎ࢊ࢏࢙ is computed as [1];
࣎ࢊ࢏࢙ ൌ ࡷ · ࢙ࢍ࢔ሺࡿሻ (25)
by replace the formulation (25) in (23) the control output can be written as;
࣎ ൌ ࣎ࢋࢗ ൅ ࡷ. ࢙ࢍ࢔ሺࡿሻ (26)
By (26) and (24) the sliding mode control of PUMA 560 robot manipulator is calculated as;
࣎ ൌ ൣࡹି૚ሺ࡮ ൅ ࡯ ൅ ࡳሻ ൅ ࡿሶ൧ࡹ ൅ ࡷ · ࢙ࢍ࢔ሺࡿሻ (27)
where ܵ ൌ ߣ݁ ൅ ݁ሶ in PD-SMC and ܵ ൌ ߣ݁ ൅ ݁ሶ ൅ ሺ
ఒ
ଶ
ሻଶ ∑ ݁ in PID-SMC.
Proof of Stability: the lyapunov formulation can be written as follows,
ࢂ ൌ
૚
૛
ࡿࢀ
. ࡹ. ࡿ
(28)
the derivation of ܸ can be determined as,
ࢂሶ ൌ
૚
૛
ࡿࢀ
. ࡹሶ . ࡿ ൅ ࡿࢀ
ࡹࡿሶ (29)
the dynamic equation of IC engine can be written based on the sliding surface as

ࡹࡿሶ ൌ െࢂࡿ ൅ ࡹࡿሶ ൅ ࡮ ൅ ࡯ ൅ ࡳ (30)
it is assumed that
ࡿࢀ൫ࡹሶ െ ૛࡮ ൅ ࡯ ൅ ࡳ൯ࡿ ൌ ૙ (31)
by substituting (30) in (29)
ࢂሶ ൌ
૚
૛
ࡿࢀ
ࡹሶ ࡿ െ ࡿࢀ
࡮ ൅ ࡯ࡿ ൅ ࡿࢀ൫ࡹࡿሶ ൅ ࡮ ൅ ࡯ࡿ ൅ ࡳ൯ ൌ ࡿࢀ൫ࡹࡿሶ ൅ ࡮ ൅ ࡯ࡿ ൅ ࡳ൯
(32)
suppose the control input is written as follows
ࢁ෡ ൌ ࢁࡺ࢕࢔࢒ଙ࢔ࢋࢇ࢘
෣ ൅ ࢁࢊଙ࢙
෣ ൌ ൣࡹି૚෣ሺ࡮ ൅ ࡯ ൅ ࡳሻ ൅ ࡿሶ൧ࡹ෡ ൅ ࡷ. ࢙ࢍ࢔ሺࡿሻ ൅ ࡮ ൅ ࡯ࡿ ൅ ࡳ (33)
by replacing the equation (33) in (32)
ࢂሶ ൌ ࡿࢀ
ሺࡹࡿሶ ൅ ࡮ ൅ ࡯ ൅ ࡳ െ ࡹ෡ ࡿሶ െ ࡮ ൅ ࡯෣ ࡿ ൅ ࡳ െ ࡷ࢙ࢍ࢔ሺࡿሻ ൌ ࡿࢀ ቀࡹ෩ ࡿሶ ൅ ࡮ ൅ ࡯෫ ࡿ ൅ ࡳ െ
ࡷ࢙ࢍ࢔ሺࡿሻ൯
(34)
it is obvious that
หࡹ෩ ࡿሶ ൅ ࡮ ൅ ࡯෫ ࡿ ൅ ࡳห ൑ หࡹ෩ ࡿሶห ൅ ห࡮ ൅ ࡯෫ ࡿ ൅ ࡳห (35)
the Lemma equation in robot arm system can be written as follows
ࡷ࢛ ൌ ൣหࡹ෩ ࡿሶห ൅ |࡮ ൅ ࡯ࡿ ൅ ࡳ| ൅ ࣁ൧࢏
, ࢏ ൌ ૚, ૛, ૜, ૝, … (36)
the equation (11) can be written as
ࡷ࢛ ൒ ቚൣࡹ෩ ࡿሶ ൅ ࡮ ൅ ࡯ࡿ ൅ ࡳ൧࢏
ቚ ൅ ࣁ࢏
(37)
therefore, it can be shown that
ࢂሶ ൑ െ ෍ ࣁ࢏
࢔
࢏ୀ૚
|ࡿ࢏|
(38)
Consequently the equation (38) guaranties the stability of the Lyapunov equation. Figure 1 is
shown pure sliding mode controller, applied to robot arm.
FIGURE 1: Block diagram of a sliding mode controller: applied to robot arm

3. METHODOLOGY: ROBUST BASELINE ON-LINE TUNING FOR STABLE
SLIDING MODE CONTROLLER
Sliding mode controller has difficulty in handling unstructured model uncertainties. It is possible to
solve this problem by combining sliding mode controller and baseline error-based tuning method
which this method can helps to eliminate the chattering in presence of switching function method
and improves the system’s tracking performance by online tuning method. In this research the
nonlinear equivalent dynamic (equivalent part) formulation problem in uncertain system is solved
by using on-line linear error-based tuning theorem. In this method linear error-based theorem is
applied to sliding mode controller to adjust the sliding surface slope. Sliding mode controller has
difficulty in handling unstructured model uncertainties and this controller’s performance is
sensitive to sliding surface slope coefficient. It is possible to solve above challenge by combining
linear error-based tuning method and sliding mode controller which this methodology can help to
improve system’s tracking performance by on-line tuning (baseline performance based tuning)
method. Based on above discussion, compute the best value of sliding surface slope coefficient
has played important role to improve system’s tracking performance especially when the system
parameters are unknown or uncertain. This problem is solved by tuning the surface slope
coefficient (ࣅ) of the sliding mode controller continuously in real-time. In this methodology, the
system’s performance is improved with respect to the pure sliding mode controller. Figure 2
shows the baseline error-based tuning sliding mode controller. Based on (23) and (27) to adjust
the sliding surface slope coefficient we define ݂መሺ‫ߣ|ݔ‬ሻ as the fuzzy based tuning.
FIGURE 2: Block diagram of a linear error-based sliding mode controller: applied to robot arm
ࢌ෠ሺ࢞|ࣅሻ ൌ ࣅࢀ
ࢾ (39)
If minimum error (ࣅ‫כ‬
) is defined by;
ࣅ‫כ‬
ൌ ࢇ࢘ࢍ ࢓࢏࢔ ሾቀࡿ࢛࢖ቚࢌ෠ሺ࢞|ࣅሻ െ ࢌሺ࢞ሻቁሿ (40)
Where ߣ்
is adjusted by an adaption law and this law is designed to minimize the error’s
parameters of ࣅ െ ࣅ‫כ‬
. adaption law in linear error-based tuning sliding mode controller is used to
adjust the sliding surface slope coefficient. Linear error-based tuning part is a supervisory
controller based on the following formulation methodology. This controller has three inputs
namely; error ሺ݁ሻ, change of error (݁ሶ) and the integral of error (∑݁) and an output namely; gain
updating factorሺߜሻ. As a summary design a linear error-based tuning is based on the following
formulation:
ࢾ ൌ ሺࡷ. ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
∑ࢋሻ ൈ ሺ ࡷ. ࢋ ൅
ሺࡷሻ૛
૛
∑ࢋ ) (41)

ࡿ࢕࢔ି࢒࢏࢔ࢋ ൌ ࢾ. ࣅࢋ ൅ ࢋሶ ֜ ࡿ࢕࢔ି࢒࢏࢔ࢋ
ൌ ሼቆࡷ. ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
෍ ࢋቇ ൈ ቆࡷ. ࢋ ൅
ሺࡷሻ૛
૛
෍ ࢋቇሽࣅࢋ ൅ ࢋሶ
ࣅࢀ࢛࢔ࢋ ൌ ࣅ. ࢾ ֜ ࣅࢀ࢛࢔ࢋ ൌ ࣅሼቆࡷ.ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽ
Where ሺߜሻ is gain updating factor, (∑ ݁) is the integral of error, (݁ሶ) is change of error, ሺ݁ሻ is error
and K is a coefficient.
Proof of Stability: The Lyapunov function in this design is defined as
ࢂ ൌ
૚
૛
ࡿࢀ
ࡹࡿ ൅
૚
૛
෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘࢀ
. ࣘ࢐
(42)
where ߛ௦௝ is a positive coefficient, ࣘ ൌ ࣅ‫כ‬
െ ࣅ, ࣂ‫כ‬
is minimum error and ߣ is adjustable parameter.
Since ‫ܯ‬ሶ െ 2ܸ is skew-symetric matrix;
ࡿࢀ
ࡹࡿሶ ൅
૚
૛
ࡿࢀ
ࡹሶ ࡿ ൌ ࡿࢀ
ሺࡹࡿሶ ൅ ࢂࡿሻ
(43)
If the dynamic formulation of robot manipulator defined by
࣎ ൌ ࡹሺࢗሻࢗሷ ൅ ࢂሺࢗ, ࢗሶ ሻࢗሶ ൅ ࡳሺࢗሻ (44)
the controller formulation is defined by
࣎ ൌ ࡹ෡ ࢗሷ ࢘ ൅ ࢂ෡ࢗሶ ࢘ ൅ ࡳ෡ െ ࣅࡿ െ ࡷ (45)
According to (43) and (44)
ࡹሺࢗሻࢗሷ ൅ ࢂሺࢗ, ࢗሶ ሻࢗሶ ൅ ࡳሺࢗሻ ൌ ࡹ෡ ࢗሷ ࢘ ൅ ࢂ෡ࢗሶ ࢘ ൅ ࡳ෡ െ ࣅࡿ െ ࡷ (46)
Since ࢗሶ ࢘ ൌ ࢗሶ െ ࡿ and ࢗሷ ࢘ ൌ ࢗሷ െ ࡿሶ
ࡹࡿሶ ൅ ሺࢂ ൅ ࣅሻࡿ ൌ ∆ࢌ െ ࡷ (47)
ࡹࡿሶ ൌ ઢࢌ െ ࡷ െ ࢂࡿ െ ࣅࡿ
The derivation of V is defined
ࢂሶ ൌ ࡿࢀ
ࡹࡿሶ ൅
૚
૛
ࡿࢀ
ࡹሶ ࡿ ൅ ෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘࢀ
. ࣘሶ ࢐
(48)
ࢂሶ ൌ ࡿࢀ
ሺࡹࡿሶ ൅ ࢂࡿሻ ൅ ෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘࢀ
. ࣘሶ ࢐
Based on (46) and (47)
ࢂሶ ൌ ࡿࢀ
ሺઢࢌ െ ࡷ െ ࢂࡿ െ ࣅࡿ ൅ ࢂࡿሻ ൅ ෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘࢀ
. ࣘሶ ࢐
(49)
where ∆݂ ൌ ሾ‫ܯ‬ሺ‫ݍ‬ሻ‫ݍ‬ሷ ൅ ܸሺ‫,ݍ‬ ‫ݍ‬ሶሻ‫ݍ‬ሶ ൅ ‫ܩ‬ሺ‫ݍ‬ሻሿ െ ∑ ࣅࢀெ
௟ୀଵ ࢾ
ࢂሶ ൌ ෍ൣࡿ࢐ሺઢࢌ‫ܒ‬ െ ࡷ࢐ሻ൧
ࡹ
ࡶୀ૚
െࡿࢀ
ࣅࡿ ൅ ෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘࢀ
. ࣘሶ ࢐
suppose ߙ is defined as follows
ࢾ࢐ ൌ ሼቆࡷ. ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽ
(50)

according to 48 and 49;
ࢂሶ ൌ ෍ ቈࡿ࢐ሺઢࢌ‫ܒ‬ െ ࣅࢀ
ࡷ.ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
෍ ࢋሿ቉
ࡹ
ࡶୀ૚
െࡿࢀ
ࣅࡿ ൅ ෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘࢀ
. ࣘሶ ࢐
(51)
Based on ࣘ ൌ ࣂ‫כ‬
െ ࣂ ՜ ࣂ ൌ ࣂ‫כ‬
െ ࣘ
ࢂሶ ൌ ෍ ቈࡿ࢐ሺઢࢌ‫ܒ‬ െ ࣂ‫ࢀכ‬
ሾሼቆࡷ. ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽሿ ൅ ࣘࢀ
ሾࢾ
ࡹ
ࡶୀ૚
ൌ ሼቆࡷ. ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽሿ቉ െࡿࢀ
ࣅࡿ
൅ ෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘࢀ
. ࣘሶ ࢐
(52)
ࢂሶ ൌ ෍ ቈࡿ࢐ሺઢࢌ‫ܒ‬ െ ሺࣅ‫כ‬
ሻࢀ
ሼቆࡷ. ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽሿ቉
ࡹ
ࡶୀ૚
െࡿࢀ
ࣅࡿ
൅ ෍
૚
ࢽ࢙࢐
ࡹ
ࡶୀ૚
ࣘ࢐
ࢀ
ሾሼቆࡷ. ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽ ൅ ࣘሶ ࢐ሿሻ
where ࣂሶ ࢐ ൌ ሼቀࡷ.ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
∑ ࢋቁ ൈ ቀࡷ. ࢋ ൅
ሺࡷሻ૛
૛
∑ ࢋቁሽ is adaption law, ‫׎‬ଚ
ሶ ൌ െࣂሶ ࢐ ൌ ሾሼቀࡷ.ࢋ ൅ ࢋሶ ൅
ሺࡷሻ૛
૛
∑ࢋቁ ൈ ቀࡷ. ࢋ ൅
ሺࡷሻ૛
૛
∑ࢋቁሽ ࢂሶ is considered by
ࢂሶ ൌ ෍ሾࡿ࢐
࢓
࢐ୀ૚
∆ࢌ࢐ െ ቆሺࣅ࢐
‫כ‬
ሻࢀ
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽቇሿ െ ࡿࢀ
ࣅࡿ
(53)
The minimum error is defined by
ࢋ࢓࢐ ൌ ∆ࢌ࢐ െ ቆሺࣅ࢐
‫כ‬
ሻࢀ
ሺࡷሻ૛
૛
ሺࡷሻ૛
૛
෍ ࢋቇሽቇ
(54)
Therefore ࢂሶ is computed as
ࢂሶ ൌ ෍ሾࡿ࢐
࢓
࢐ୀ૚
ࢋ࢓࢐ሿ െ ࡿࢀ
ࣅࡿ
(55)
൑ ∑ |ࡿ࢐
࢓
࢐ୀ૚ ||ࢋ࢓࢐| െ ࡿࢀ
ࣅࡿ
ൌ ෍ |ࡿ࢐
࢓
࢐ୀ૚
||ࢋ࢓࢐| െ ࣅ࢐ࡿ࢐
૛
ൌ ෍ |ࡿ࢐
࢓
࢐ୀ૚
|൫หࢋ࢓࢐ห െ ࣅ࢐ࡿ࢐൯
(56)
4. RESULTS
This part is focused on compare between Sliding Mode Controller (SMC) and baseline error-
based tuning Sliding Mode Controller (LTSMC). These controllers were tested by step
responses. In this simulation, to control position of PUMA robot manipulator the first, second, and
third joints are moved from home to final position without and with external disturbance. The
simulation was implemented in Matlab/Simulink environment.
Trajectory performance, torque performance, disturbance rejection, steady state error and
RMS error are compared in these controllers. These systems are tested by band limited white
noise with a predefined 40% of relative to the input signal amplitude. This type of noise is used to

external disturbance in continuous and hybrid systems and applied to nonlinear dynamic of these
controllers.
Tracking performances: In sliding mode controller; controllers performance are depended on
the gain updating factor (‫)ܭ‬ and sliding surface slope coefficient (ߣ). These two coefficients are
computed by trial and error in SMC. The best possible coefficients in step SMC are; ߣଵ ൌ 1 , ߣଶ ൌ
6, ߣଷ ൌ 8; ‫ܭ‬௣ ൌ ‫ܭ‬௩ ൌ ‫ܭ‬௜ ൌ 10; ‫׎‬ଵ ൌ ‫׎‬ଶ ൌ ‫׎‬ଷ ൌ 0.1. In linear error-based tuning sliding mode
controller the sliding surface gain is adjusted online depending on the last values of error ሺ݁ሻ,
change of error (݁ሶ) and the integral of error (∑݁) by sliding surface slope updating factor (ߜሻ.
Figure 3 shows tracking performance in baseline error-based tuning sliding mode controller
(LTSMC) and sliding mode controller (SMC) without disturbance for step trajectory.
FIGURE 3: LTSMC and SMC for first, second and third link step trajectory performance without disturbance
Based on Figure 3 it is observed that, the overshoot in LTSMC is 0% and in SMC’s is 1%, and the
rise time in LTSMC’s is 0.48 seconds and in SMC’s is 0.4 second. From the trajectory MATLAB
simulation for LTSMC and SMC in certain system, it was seen that all of two controllers have
acceptable performance.
Disturbance Rejection: Figure 4 shows the power disturbance elimination in LTSMC and SMC
with disturbance for step trajectory. The disturbance rejection is used to test the robustness
comparisons in these two controllers for step trajectory. A band limited white noise with
predefined of 40% the power of input signal value is applied to the step trajectory. It found fairly
fluctuations in SMC trajectory responses.

FIGURE 4: LTSMC and SMC for first, second and third link trajectory with 40%external disturbance: step
trajectory
Based on Figure 4; by comparing step response trajectory with 40% disturbance of relative to the
input signal amplitude in LTSMC and SMC, LTSMC’s overshoot about (0%) is lower than SMC’s
(8%). SMC’s rise time (0.5 seconds) is lower than LTSMC’s (0.8 second). Besides the Steady
State and RMS error in LTSMC and SMC it is observed that, error performances in LTSMC
(Steady State error =1.3e-12 and RMS error=1.8e-12) are about lower than SMC’s (Steady
State error=10e-4 and RMS error=11e-4). Based on Figure 4, SMC has moderately oscillation in
trajectory response with regard to 40% of the input signal amplitude disturbance but LTSMC has
stability in trajectory responses in presence of uncertainty and external disturbance. Based on
Figure 4 in presence of 40% unstructured disturbance, LTSMC’s is more robust than SMC
because LTSMC can auto-tune the sliding surface slope coefficient as the dynamic manipulator
parameter’s change and in presence of external disturbance whereas SMC cannot.
Torque Performance: Figure 5 and 6 have indicated the power of chattering rejection in LTSMC
and SMC with 40% disturbance and without disturbance.

FIGURE 5: LTSMC and SMC for first, second and third link torque performance without disturbance
Figure 5 shows torque performance for first three links robot manipulator in LTSMC and SMC
without disturbance. Based on Figure 5, LTSMC and SMC give considerable torque performance
in certain system and all two controllers eliminate the chattering phenomenon in certain system.
Figure 6 has indicated the robustness in torque performance for three links robot manipulator in
LTSMC and SMC in presence of 40% disturbance. Based on Figure 6, it is observed that SMC
controller has oscillation but LTSMC has steady in torque performance. This is mainly because
pure SMC are robust but they have limitation in presence of external disturbance. The LTSMC
gives significant chattering elimination when compared to SMC. This elimination of chattering
phenomenon is very significant in presence of 40% disturbance. This challenge is one of the most
important objectives in this research.

FIGURE 6: LTSMC and SMC for first, second and third link torque performance with40% disturbance
SMC has limitation to eliminate the chattering in presence of highly external disturbance (e.g.,
40% disturbance) but LTSMC is a robust against to highly external disturbance.
Steady state error: Figure 7 is shown the error performance in LTSMC and SMC for three links
robot manipulator. The error performance is used to test the disturbance effect comparisons of
these controllers for step trajectory. All three joint’s inputs are step function with the same step
time (step time= 1 second), the same initial value (initial value=0) and the same final value (final
value=5). Based on Figure 3, LTSMC’s rise time is about 0.48 second and SMC’s rise time is
about 0.4 second which caused to create a needle wave in the range of 5 (amplitude=5) and the
different width. In this system this time is transient time and this part of error introduced as a
transient error. Besides the Steady State and RMS error in LTSMC and SMC it is observed that,
error performances in LTSMC (Steady State error =1.8e-10 and RMS error=1.16e-12) are
about lower than SMC’s (Steady State error=1e-8 and RMS error=1.2e-6).

FIGURE 7: LTSMC and SMC for first, second and third link steady state error without disturbance: step
trajectory
The LTSMC gives significant steady state error performance when compared to SMC. When
applied 40% disturbances in LTSMC the RMS error increased approximately 0.0164% (percent of
increase the LTSMC RMS error=
ሺସ଴% ௗ௜௦௧௨௥௕௔௡௖௘ ோெௌ ௘௥௥௢௥
௡௢ ௗ௜௦௧௨௥௕௔௡௖௘ோெௌ ௘௥௥௢௥
ൌ
ଵ.ଵ଺௘ିଵଶ
ଵ.ଵ௘ିଵଶ
ൌ 0.0122%) and in SMC the
RMS error increased approximately 9.17% (percent of increase the PD-SMC RMS
error=
ሺସ଴% ௗ௜௦௧௨௥௕௔௡௖௘ ோெௌ ௘௥௥௢௥
௡௢ ௗ௜௦௧௨௥௕௔௡௖௘ ோெௌ ௘௥௥௢௥
ൌ
ଵଵ௘ିସ
ଵ.ଶ௘ି଺
ൌ 9.17%). In this part LTSMC and SMC have been
comparatively evaluation through MATLAB simulation, for 3DOF robot manipulator control. It is
observed that however LTSMC is dependent of nonlinear dynamic equation of robot manipulator
but it can guarantee the trajectory following and eliminate the chattering phenomenon in certain
systems, structure uncertain systems and unstructured model uncertainties by online tuning
method.
5. CONCLUSION
In this research, a baseline error-based tuning sliding mode controller (LTSMC) is design and
applied to robot manipulator. Pure sliding mode controller has difficulty in handling unstructured
model uncertainties. It is possible to solve this problem by combining sliding mode controller and
linear error-based tuning. The sliding surface gain (ߣ) is adjusted by linear error-based tuning
method. The sliding surface slope updating factor (ߜ) of linear error-based tuning part can be
changed with the changes in error, change of error and the integral (summation) of error. Sliding
surface gain is adapted on-line by sliding surface slope updating factor. In pure sliding mode
controller the sliding surface gain is chosen by trial and error, which means pure sliding mode
controller had to have a prior knowledge of the system uncertainty. If the knowledge is not
available error performance and chattering phenomenon are go up. In linear error-based tuning
sliding mode controller the sliding surface gain are updated on-line to compensate the system
unstructured uncertainty. The simulation results exhibit that the linear error-based tuning sliding
mode controller works well in various situations.

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Robust Sliding Surface Slope Adjustment in Sliding Mode Control for Robot Manipulator,”
World Applied Science Journal, 12 (12): 2330-2336, 2011.

[35] Farzin Piltan , A. H. Aryanfar, Nasri B. Sulaiman, M. H. Marhaban and R. Ramli “Design
Adaptive Fuzzy Robust Controllers for Robot Manipulator,” World Applied Science
Journal, 12 (12): 2317-2329, 2011.
[36] Farzin Piltan, N. Sulaiman , Arash Zargari, Mohammad Keshavarz, Ali Badri , “Design PID-
Like Fuzzy Controller With Minimum Rule Base and Mathematical Proposed On-line
Tunable Gain: Applied to Robot Manipulator,” International Journal of Artificial intelligence
and expert system, 2 (4):184-195, 2011.
[37] Farzin Piltan, Nasri Sulaiman, M. H. Marhaban and R. Ramli, “Design On-Line Tunable
Gain Artificial Nonlinear Controller,” Journal of Advances In Computer Research, 2 (4): 75-
83, 2011.
[38] Farzin Piltan, N. Sulaiman, Payman Ferdosali, Iraj Assadi Talooki, “ Design Model Free
Fuzzy Sliding Mode Control: Applied to Internal Combustion Engine,” International Journal
of Engineering, 5 (4):302-312, 2011.
[39] Farzin Piltan, N. Sulaiman, Samaneh Roosta, M.H. Marhaban, R. Ramli, “Design a New
Sliding Mode Adaptive Hybrid Fuzzy Controller,” Journal of Advanced Science &
Engineering Research , 1 (1): 115-123, 2011.
[40] Farzin Piltan, Atefe Gavahian, N. Sulaiman, M.H. Marhaban, R. Ramli, “Novel Sliding Mode
Controller for robot manipulator using FPGA,” Journal of Advanced Science & Engineering
Research, 1 (1): 1-22, 2011.
[41] Farzin Piltan, N. Sulaiman, A. Jalali & F. Danesh Narouei, “Design of Model Free Adaptive
Fuzzy Computed Torque Controller: Applied to Nonlinear Second Order System,”
International Journal of Robotics and Automation, 2 (4):232-244, 2011.
[42] Farzin Piltan, N. Sulaiman, Iraj Asadi Talooki, Payman Ferdosali, “Control of IC Engine:
Design a Novel MIMO Fuzzy Backstepping Adaptive Based Fuzzy Estimator Variable
Structure Control ,” International Journal of Robotics and Automation, 2 (5):360-380, 2011.
[43] Farzin Piltan, N. Sulaiman, Payman Ferdosali, Mehdi Rashidi, Zahra Tajpeikar, “Adaptive
MIMO Fuzzy Compensate Fuzzy Sliding Mode Algorithm: Applied to Second Order
Nonlinear System,” International Journal of Engineering, 5 (5): 380-398, 2011.
[44] Farzin Piltan, N. Sulaiman, Hajar Nasiri, Sadeq Allahdadi, Mohammad A. Bairami, “Novel
Robot Manipulator Adaptive Artificial Control: Design a Novel SISO Adaptive Fuzzy Sliding
Algorithm Inverse Dynamic Like Method,” International Journal of Engineering, 5 (5): 399-
418, 2011.
[45] Farzin Piltan, N. Sulaiman, Sadeq Allahdadi, Mohammadali Dialame, Abbas Zare, “Position
Control of Robot Manipulator: Design a Novel SISO Adaptive Sliding Mode Fuzzy PD Fuzzy
Sliding Mode Control,” International Journal of Artificial intelligence and Expert System, 2
(5):208-228, 2011.
[46] Farzin Piltan, SH. Tayebi HAGHIGHI, N. Sulaiman, Iman Nazari, Sobhan Siamak, “Artificial
Control of PUMA Robot Manipulator: A-Review of Fuzzy Inference Engine And Application
to Classical Controller ,” International Journal of Robotics and Automation, 2 (5):401-425,
2011.
[47] Farzin Piltan, N. Sulaiman, Abbas Zare, Sadeq Allahdadi, Mohammadali Dialame, “Design
Adaptive Fuzzy Inference Sliding Mode Algorithm: Applied to Robot Arm,” International
Journal of Robotics and Automation , 2 (5): 283-297, 2011.

[48] Farzin Piltan, Amin Jalali, N. Sulaiman, Atefeh Gavahian, Sobhan Siamak, “Novel Artificial
Control of Nonlinear Uncertain System: Design a Novel Modified PSO SISO Lyapunov
Based Fuzzy Sliding Mode Algorithm ,” International Journal of Robotics and Automation, 2
(5): 298-316, 2011.
[49] Farzin Piltan, N. Sulaiman, Amin Jalali, Koorosh Aslansefat, “Evolutionary Design of
Mathematical tunable FPGA Based MIMO Fuzzy Estimator Sliding Mode Based Lyapunov
Algorithm: Applied to Robot Manipulator,” International Journal of Robotics and Automation,
2 (5):317-343, 2011.
[50] Farzin Piltan, N. Sulaiman, Samaneh Roosta, Atefeh Gavahian, Samira Soltani,
“Evolutionary Design of Backstepping Artificial Sliding Mode Based Position Algorithm:
Applied to Robot Manipulator,” International Journal of Engineering, 5 (5):419-434, 2011.
[51] Farzin Piltan, N. Sulaiman, S.Soltani, M. H. Marhaban & R. Ramli, “An Adaptive sliding
surface slope adjustment in PD Sliding Mode Fuzzy Control for Robot Manipulator,”
International Journal of Control and Automation , 4 (3): 65-76, 2011.
[52] Farzin Piltan, N. Sulaiman, Mehdi Rashidi, Zahra Tajpaikar, Payman Ferdosali, “Design
and Implementation of Sliding Mode Algorithm: Applied to Robot Manipulator-A Review ,”
International Journal of Robotics and Automation, 2 (5):265-282, 2011.
[53] Farzin Piltan, N. Sulaiman, Amin Jalali, Sobhan Siamak, and Iman Nazari, “Control of
Robot Manipulator: Design a Novel Tuning MIMO Fuzzy Backstepping Adaptive Based
Fuzzy Estimator Variable Structure Control ,” International Journal of Control and
Automation, 4 (4):91-110, 2011.
[54] Farzin Piltan, N. Sulaiman, Atefeh Gavahian, Samaneh Roosta, Samira Soltani, “On line
Tuning Premise and Consequence FIS: Design Fuzzy Adaptive Fuzzy Sliding Mode
Controller Based on Lyaponuv Theory,” International Journal of Robotics and Automation, 2
(5):381-400, 2011.
[55] Farzin Piltan, N. Sulaiman, Samaneh Roosta, Atefeh Gavahian, Samira Soltani, “Artificial
Chattering Free on-line Fuzzy Sliding Mode Algorithm for Uncertain System: Applied in
Robot Manipulator,” International Journal of Engineering, 5 (5):360-379, 2011.
[56] Farzin Piltan, N. Sulaiman and I.AsadiTalooki, “Evolutionary Design on-line Sliding Fuzzy
Gain Scheduling Sliding Mode Algorithm: Applied to Internal Combustion Engine,”
International Journal of Engineering Science and Technology, 3 (10):7301-7308, 2011.
[57] Farzin Piltan, Nasri B Sulaiman, Iraj Asadi Talooki and Payman Ferdosali.,” Designing On-
Line Tunable Gain Fuzzy Sliding Mode Controller Using Sliding Mode Fuzzy Algorithm:
Applied to Internal Combustion Engine,” world applied science journal (WASJ), 15 (3): 422-
428, 2011
[58] B. K. Yoo and W. C. Ham, "Adaptive control of robot manipulator using fuzzy
compensator," Fuzzy Systems, IEEE Transactions on, No. 2, vol. 8, pp. 186-199, 2002.
[59] H. Medhaffar, N. Derbel and T. Damak, "A decoupled fuzzy indirect adaptive sliding mode
controller with application to robot manipulator," International Journal of Modelling,
Identification and Control, No. 1, vol. 1, pp. 23-29, 2006.
[60] Y. Guo and P. Y. Woo, "An adaptive fuzzy sliding mode controller for robotic manipulators,"
Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on, No.
2, vol. 33, pp. 149-159, 2003.

[61] C. M. Lin and C. F. Hsu, "Adaptive fuzzy sliding-mode control for induction servomotor
systems," Energy Conversion, IEEE Transactions on, No. 2, vol. 19, pp. 362-368, 2004.
[62] Xiaosong. Lu, "An investigation of adaptive fuzzy sliding mode control for robot
manipulator," Carleton university Ottawa,2007.
[63] S. Lentijo, S. Pytel, A. Monti, J. Hudgins, E. Santi and G. Simin, "FPGA based sliding mode
control for high frequency power converters," IEEE Conference, 2004, pp. 3588-3592.
[64] B. S. R. Armstrong, "Dynamics for robot control: friction modeling and ensuring excitation
during parameter identification," 1988.
[65] C. L. Clover, "Control system design for robots used in simulating dynamic force and
moment interaction in virtual reality applications," 1996.
[66] K. R. Horspool, Cartesian-space Adaptive Control for Dual-arm Force Control Using
Industrial Robots: University of New Mexico, 2003.
[67] B. Armstrong, O. Khatib and J. Burdick, "The explicit dynamic model and inertial
parameters of the PUMA 560 arm," IEEE Conference, 2002, pp. 510-518.
[68] P. I. Corke and B. Armstrong-Helouvry, "A search for consensus among model parameters
reported for the PUMA 560 robot," IEEE Conference, 2002, pp. 1608-1613.
[69] Farzin Piltan, N. Sulaiman, M. H. Marhaban, Adel Nowzary, Mostafa Tohidian,” “Design of
FPGA based sliding mode controller for robot manipulator,” International Journal of
Robotic and Automation, 2 (3): 183-204, 2011.
[70] I. Eksin, M. Guzelkaya and S. Tokat, "Sliding surface slope adjustment in fuzzy sliding
mode controller," Mediterranean Conference, 2002, pp. 160-168.
[71] Farzin Piltan, H. Rezaie, B. Boroomand, Arman Jahed,” Design robust back stepping
online tuning feedback linearization control applied to IC engine,” International Journal of
Advance Science and Technology, 42: 183-204, 2012.
[72] Farzin Piltan, I. Nazari, S. Siamak, P. Ferdosali ,”Methodology of FPGA-based
mathematical error-based tuning sliding mode controller” International Journal of Control
and Automation, 5(1): 89-110, 2012.
[73] Farzin Piltan, M. A. Dialame, A. Zare, A. Badri ,”Design Novel Lookup table changed Auto
Tuning FSMC: Applied to Robot Manipulator” International Journal of Engineering, 6(1):
25-40, 2012.
[74] Farzin Piltan, B. Boroomand, A. Jahed, H. Rezaie ,”Methodology of Mathematical Error-
Based Tuning Sliding Mode Controller” International Journal of Engineering, 6(2): 96-112,
2012.
[75] Farzin Piltan, F. Aghayari, M. R. Rashidian, M. Shamsodini, ”A New Estimate Sliding Mode
Fuzzy Controller for Robotic Manipulator” International Journal of Robotics and
Automation, 3(1): 45-58, 2012.
[76] Farzin Piltan, M. Keshavarz, A. Badri, A. Zargari , ”Design novel nonlinear controller
applied to robot manipulator: design new feedback linearization fuzzy controller with
minimum rule base tuning method” International Journal of Robotics and Automation,
3(1): 1-18, 2012.

[77] Piltan, F., et al. "Design sliding mode controller for robot manipulator with artificial tunable
gain". Canaidian Journal of pure and applied science, 5 (2), 1573-1579, 2011.
[78] Samira Soltani & Farzin Piltan, “Design Artificial Nonlinear Controller Based on Computed
Torque like Controller with Tunable Gain”. World Applied Science Journal,14 (9): 1306-
1312, 2011.
[79] Piltan, F., et al. "Design sliding mode controller for robot manipulator with artificial tunable
gain". Canaidian Journal of pure and applied science, 5 (2), 1573-1579, 2011.
[80] Farzin Piltan, A. Hosainpour, E. Mazlomian, M.Shamsodini, M.H Yarmahmoudi. ”Online
Tuning Chattering Free Sliding Mode Fuzzy Control Design: Lyapunov Approach”
International Journal of Robotics and Automation, 3(3): 2012.
[81] Farzin Piltan , M.H. Yarmahmoudi, M. Shamsodini, E.Mazlomian, A.Hosainpour. ” PUMA-
560 Robot Manipulator Position Computed Torque Control Methods Using
MATLAB/SIMULINK and Their Integration into Graduate Nonlinear Control and MATLAB
Courses” International Journal of Robotics and Automation, 3(3): 2012.
[82] Farzin Piltan, R. Bayat, F. Aghayari, B. Boroomand. “Design Error-Based Linear Model-
Free Evaluation Performance Computed Torque Controller” International Journal of
Robotics and Automation, 3(3): 2012.
[83] Farzin Piltan, S. Emamzadeh, Z. Hivand, F. Shahriyari & Mina Mirazaei . ” PUMA-560 Robot
Manipulator Position Sliding Mode Control Methods Using MATLAB/SIMULINK and Their
Integration into Graduate/Undergraduate Nonlinear Control, Robotics and MATLAB
Courses” International Journal of Robotics and Automation, 3(3): 2012.
[84] Farzin Piltan, S. Rahmdel, S. Mehrara, R. Bayat.” Sliding Mode Methodology Vs. Computed
Torque Methodology Using MATLAB/SIMULINK and Their Integration into Graduate
Nonlinear Control Courses” International Journal of Engineering, 3(3): 2012.
[85] Farzin Piltan, M. Mirzaie, F. Shahriyari, Iman Nazari & S. Emamzadeh.” Design Baseline
Computed Torque Controller” International Journal of Engineering, 3(3): 2012.

Evaluation Performance of 2nd Order Nonlinear System: Baseline Control Tunable Gain Sliding Mode Methodology

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